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Properties

Label 1-293-293.84-r0-0-0
Degree $1$
Conductor $293$
Sign $-0.839 - 0.543i$
Analytic cond. $1.36068$
Root an. cond. $1.36068$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.150 + 0.988i)2-s + (0.996 + 0.0859i)3-s + (−0.954 − 0.296i)4-s + (−0.798 + 0.601i)5-s + (−0.234 + 0.972i)6-s + (−0.618 + 0.785i)7-s + (0.436 − 0.899i)8-s + (0.985 + 0.171i)9-s + (−0.474 − 0.880i)10-s + (−0.999 − 0.0430i)11-s + (−0.925 − 0.377i)12-s + (−0.890 − 0.455i)13-s + (−0.683 − 0.729i)14-s + (−0.847 + 0.530i)15-s + (0.823 + 0.566i)16-s + (−0.150 + 0.988i)17-s + ⋯
L(s)  = 1  + (−0.150 + 0.988i)2-s + (0.996 + 0.0859i)3-s + (−0.954 − 0.296i)4-s + (−0.798 + 0.601i)5-s + (−0.234 + 0.972i)6-s + (−0.618 + 0.785i)7-s + (0.436 − 0.899i)8-s + (0.985 + 0.171i)9-s + (−0.474 − 0.880i)10-s + (−0.999 − 0.0430i)11-s + (−0.925 − 0.377i)12-s + (−0.890 − 0.455i)13-s + (−0.683 − 0.729i)14-s + (−0.847 + 0.530i)15-s + (0.823 + 0.566i)16-s + (−0.150 + 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $-0.839 - 0.543i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ -0.839 - 0.543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1622842670 + 0.5490879339i\)
\(L(\frac12)\) \(\approx\) \(-0.1622842670 + 0.5490879339i\)
\(L(1)\) \(\approx\) \(0.5472384813 + 0.5515707644i\)
\(L(1)\) \(\approx\) \(0.5472384813 + 0.5515707644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (-0.150 + 0.988i)T \)
3 \( 1 + (0.996 + 0.0859i)T \)
5 \( 1 + (-0.798 + 0.601i)T \)
7 \( 1 + (-0.618 + 0.785i)T \)
11 \( 1 + (-0.999 - 0.0430i)T \)
13 \( 1 + (-0.890 - 0.455i)T \)
17 \( 1 + (-0.150 + 0.988i)T \)
19 \( 1 + (-0.847 + 0.530i)T \)
23 \( 1 + (-0.234 - 0.972i)T \)
29 \( 1 + (-0.548 + 0.835i)T \)
31 \( 1 + (-0.798 - 0.601i)T \)
37 \( 1 + (0.357 + 0.933i)T \)
41 \( 1 + (0.908 - 0.417i)T \)
43 \( 1 + (-0.976 + 0.213i)T \)
47 \( 1 + (0.192 + 0.981i)T \)
53 \( 1 + (-0.999 - 0.0430i)T \)
59 \( 1 + (0.714 - 0.699i)T \)
61 \( 1 + (0.512 + 0.858i)T \)
67 \( 1 + (0.0215 + 0.999i)T \)
71 \( 1 + (-0.991 + 0.128i)T \)
73 \( 1 + (-0.954 + 0.296i)T \)
79 \( 1 + (0.512 + 0.858i)T \)
83 \( 1 + (0.966 + 0.255i)T \)
89 \( 1 + (0.512 - 0.858i)T \)
97 \( 1 + (-0.0645 + 0.997i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.0267907528284815947677227083, −23.73508949358949958882037624573, −23.2956624538566175462459215180, −21.94231552663465414723108960179, −20.99014554864597459617020295906, −20.255065592582969334688933213841, −19.58040888901883047146015626806, −19.07180207217444304683996194588, −17.91655357873640412963995573482, −16.68061826759447020458994846124, −15.75239592587427551648885299757, −14.58449783893869823137077606502, −13.40078744946603070261731843258, −12.99493669039896440859206381433, −11.95425087764949320411663519109, −10.78865801393872802526957501899, −9.69274707107340533083260266257, −9.054466366868438670992197153497, −7.837489216476213440497024901614, −7.257967833076170385323602503612, −4.94500105288141369121548509241, −4.08748569240541129357321003320, −3.13647630377089302299488120985, −2.02164867497926804960527851773, −0.335174630562237670840282574843, 2.37844625159187782988105929448, 3.48276649973170354873915437379, 4.596476806976843067176063097486, 5.97845653647291552079777089015, 7.09048787654648755578640117753, 8.00481584520555194961253309500, 8.601309098985166047136138618218, 9.8371086541642335547904094217, 10.60765174237962849583864932789, 12.58237875685616969297297674310, 13.00466087069787108126373794257, 14.638160355016348620231306976457, 14.854599590623171337077977255564, 15.72499933053151814842979019844, 16.49259599571254258915088484900, 17.99017938494202967788679904852, 18.8969696703593659072807110544, 19.218573440278688487770032735600, 20.39534647370443861610106146855, 21.83507544567423183842820193566, 22.352410600039142820665078636415, 23.59540086691115488708875299077, 24.23956177315792802777285008309, 25.32320154838010813291813001247, 25.93600369529142077841533165639

Graph of the $Z$-function along the critical line